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Fig. 18.3. Spatial likelihood-ratio processor for Gauss—Markov (spatial) possibility distribution of noise 
of Example III. Dotted lines enclose processing due to spatially correlated noise. 
The first two additive terms on the right of Eq. (15) are signal-bias terms. The 
next three additive terms are those for normal pattern formation [see Example 
I, Eq. (11)]; however, the coefficients of the pattern terms for inputs ,x, and 
mx, differ from the coefficient (1+ py) for the majority of the inputs. Finally, 
additional processing of the inputs beyond pattern formation is indicated by the 
last additive term. The spatial likelihood-ratio processor design for this ex- 
ample is drawn in Fig. 18.3. 
Comparing Figs. 18.2 and 18.3, we see thatthe essential difference in spatial 
processing due to spatially correlated noise (Example II) versus independent 
noise (Example I) is the multiplication of an input by both the succeeding and 
following values of the signal. For a plane wavefront, this operation would con- 
sist of delaying and advancing an input fromthe ith hydrophone as though it were 
received at the (s;-1)th and(@+1)th hydrophones, respectively. If we allow the 
noise to become spatially independent, p, goes to zero and the Markovian proc- 
essing vanishes. 
Example III: The previous example may be generalized to include first-order 
Gauss—Markov perturbation of the signal wavefront about its ideal mean, in 
addition to first-order Gauss—Markov noise in space. The spatial likelihood- 
ratio processor design for several interior (i 41,M) hydrophone outputs is given 
in Fig. 18.4, together with the values of the amplification coefficient in Table I. 
Comparison with Fig. 18.3 reveals two new processes: multiplication of hydro- 
phone outputs and squaring of the individual hydrophone outputs. By juggling 
values of 0,, py, o&, and of, a smooth transition from the case of classic pat- 
tern formation alone (Example I), through correlated noise processing (Ex- 
ample II), to completely incoherent spatial processing can be found. This latter 
